1. Field of the Invention
The present invention relates to a solution of an optimization problem of design parameters, and more particularly to an automatic design technique for optimization of shapes including topology of structural members.
2. Description of the Related Art
The term “structural topology optimum design” means a problem for deciding the topology and shape dimensions of a structural member, which are optimum under given conditions. Hereinafter, the topology and shape dimensions of a structural member will be referred to as “design argument functions”, and the above decision problem will be referred to as a “design-variable-function optimization problem”. The reason why the term “argument functions” is used resides in that the topology and shape dimensions are given as functions of a three-dimensional space.
In the design-variable-function optimization problem, an optimization problem of a state argument function must be solved for a value of each design argument function. From this point of view, the structural topology optimum design can be regarded as a double structure optimization problem that contains an optimization problem of a state argument function on the inner side and an optimization problem of a design argument function on the outer side. As to the optimization problem of a state argument function on the inner side, the concept of dividing a space into a finite number of elements is employed based on accumulation of known techniques.
For a problem taking strain energy of a structural member as an evaluation generic function, in particular, a finite element method is generally employed as a solution technique for that problem. A direct method with respect to a linear equation is employed for solution of the finite element method.
On the other hand, regarding the optimization problem of a design argument function, the following three kinds of methods have been primarily provided (see Reference 1, S. Bulman, J. Sienz, E. Hinton: “Comparisons between algorithms for structural topology optimization using a series of benchmark studies”, Computers and Structures, 79, pp. 1203-1218 (2001), or Reference 2, Y-L. Hsu, M-S. Hsu, C-T. Chen: “Interpreting results from topology optimization using density contours”, Computers and Structures, 79, pp. 1049-1058 (2001)):
1. Evolutionary method (referred to as “E method” below)
2. Homogenization method (referred to as “H method” below)
3. Material distribution method (referred to as “MD method” below)
According to the E method, each of partial spaces resulting from dividing a space is called a cell, and creation and disappearance of cells are repeated in accordance with proper rules. A structural member is given as a set of cells that exist finally. A definite structural member is obtained by allowing only two states represented by whether the cell exists or not. Also, because the E method does not employ differential information of the evaluation generic function and can avoid trapping to a local optimum solution, the E method is effective in the case of the evaluation generic function having multiple peaks.
Reference 3 (Japanese Patent Laid-Open No. 2001-134628) proposes an optimization design device for a skeleton structural member, which employs one kind of the E method, i.e., a genetic search method. The proposed optimization design device comprises an approximate optimization computing device using approximation formulae for discrete design variable data such as section dimensions of a skeleton structural member, and a close optimization computing device using the design variable data. Those two computing devices are combined with each other to be adapted for an actual design problem containing a large number of design variables.
The H method enables sensitivity analysis to be employed by assuming a finer structure for a structural element positioned in each of divided partial spaces and introducing a new design argument function having a continuous value. Here, the term “sensitivity analysis” means an analytical technique utilizing differential information of the evaluation generic function with respect to the design argument function. If the sensitivity analysis is enabled, it becomes possible to employ an iterative method, such as a gradient method. As a result, a computation time required for search of at least a local optimum solution can be greatly cut in comparison with a round-robin technique, such as the E method. (See Reference 4, Hiroshi Yamakawa, “Saitekika Dezain (Optimization Design)”, Computation Dynamics and CAE Series 9, Baifukan Co., Ltd. (1996).)
The MD method is a method for expressing changes in topology and shape dimensions of a structural member by assigning, to each element, a real number in the range of 0 to 1 representing the presence probability of the structural member. The MD method is similar to the H method in that the sensitivity analysis is enabled by replacing discrete information, which indicates whether the structural member is present or not, with a continuous value representing the presence probability of the structural member. However, because the number of parameters is smaller in the MD method than in the H method, the MD method is advantageous in that modeling is easier and the application range is wider.
Reference 5, Fujii, Suzuki and Otsubo, “Structure Phase Optimization Using Voxel Finite Element Method”, Transactions of JSCES, Paper No. 200000010(2000), describes a phase optimization technique for a structure based on the MD method. This technique has features given below.
(1) Because of using a voxel finite element method (in which a space is divided at equal intervals), element rigidity matrices for all elements are the same. Accordingly, by computing the element rigidity matrix once in advance, it is available in subsequent computations. Furthermore, since elements are regularly arranged, there is no need of storing node number information for each element.
(2) A pre-processed conjugate gradient method and an element-by-element method are combined with each other to solve large-scaled simultaneous linear equations. As a result, a solution can be obtained without setting up an entire rigidity matrix, and therefore the memory capacity required for processing can be reduced.
(3) A homogenization method requires 6 design variables (in the three-dimensional case) for one element. Further, each time the design variable is changed, the element rigidity matrix must be computed again. On the other hand, by employing a density method expressing the presence probability of the structural member as a density ratio, only one design variable is required for one element. In addition, changes of the design variable do not affect the element rigidity matrix.
The above-mentioned known techniques, however, have problems given below.
Generally, a structure optimization problem is formulated as a double optimization problem containing an optimization problem of a state variable vector per iterative step for an optimization problem of a design variable vector. Assuming the optimization problem of a state variable vector to be optimization on the outer side and the optimization problem of a design variable vector to be optimization on the inner side, the optimization on the inner side is a problem of determining the state variable vector with the design variable vector being a parameter, i.e., the design variable vector being fixed. This problem is usually called structural analysis and can be solved based on the finite element method by using the solution technique for a linear equation.
However, if a structure changes and a structural member does not exist in a certain area any more, for example, if a structural member is holed, the design variable of the corresponding element becomes 0 and the Young's modulus of that element also eventually becomes 0. This leads to a result that the problem cannot be solved by the direct method because a coefficient matrix of the linear equation is not fully filled with elements and an inverse matrix does not exist.
For that reason, most of the known techniques employ a method of using a value of not exactly 0, i.e., replacing 0 with a small value close to 0, when the design variable vector becomes 0. This method is similarly applied to the voxel finite element method employed in Reference 5.
Setting the design variable of the element to a small value, however, corresponds to the case in which there exists a thin film or a weak member from a physical point of view. Stated another way, an area where no materials are present is not exactly expressed in the known techniques.
Further, as seen from the figure showing the result in Reference 5, a structural element area not contributing to total strain energy, i.e., a floating island area or a projection area, is left in the result. Such a structural element area should be finally removed, but a method for removing that area is not explained in Reference 5.
In FIG. 16, by way of example, when elements 512, 513 and 514 are neither subjected to any weight nor supported, those elements have no contributions from the standpoint of strength. In other words, those elements are ones not contributing to total strain energy and hence should be removed in accordance with a design demand for minimum total weight of the structural members. However, it is difficult to find those elements so long as the solution technique is based only on simple geometrical information, i.e., on connection of characteristic functions.